Orthogonal functions, expansions terms

If population coefficients are assigned to each multipole term, rather than orbital products,23 the expansion is now in terms of orthogonal functions. [TTie functions centered on different centers are not orthogonal to each other.] For this case correlation coefficients should be very small. Except for (2px )2, (2py )2, (2pz )2, and (2s)2, the orbital product population coefficient is proportional to the one assigned to a multipole. If we assign Py to FjCcos 0), Qx to P cos 20 and Qs to P%, then... 

The familiar Fourier series is only one special form of an expansion in terms of orthogonal functions. Figure 22-1, which gives a plot of the function... 

It may be mentioned that in some cases it is convenient to make use of complete sets of functions which are not mutually orthogonal. An arbitrary function can be expanded in terms of the functions of such a set the determination of the values of the coefficients is, however, not so simple as for orthogonal functions. An example of an expansion of this type occurs in Section 24. 

In certain applications of expansions in terms of orthogonal functions, we shall obtain expressions of the form... 

So far our treatment differs from the previous discussion of non-degenerate levels only in the use of x i instead of Vhhe., in the introduction of a general expression for unperturbed functions instead of the arbitrary set ik. In the next step we likewise follow the previous treatment, in which the quantities xp k and Hn4/ k were expanded in terms of the complete set of orthogonal functions . Here, however, we must in addition express x i in terms of the set by means of Equation 24-4, in which the coefficients < are so far arbitrary. Therefore we introduce the expansions... 

If, however, we did not know this and used a linear expansion of the ground state function in terms of a set of p- and d-type functions in the variation method we would not, of course, obtain the correct ground-state solution (or even a reasonable approximation to it). This is simply because all the p functions and the d functions are orthogonal to the ground-state function by symmetry. 

A specific aspect of the ionization problem are the expansion coefficients Ci, i Ek,t) related to the electronic continua. Their dependence on the continuous variable Ek leads to a noncountably infinite set of differential equations. To arrive at a computationally manageable scheme, the continua have to be discretized. An ingenious discretization scheme, which proves particularly efficient in the present context, has been proposed by Bur key and Cantrell. The energy-dependent coefficients are expanded in terms of polynomials which are orthogonal with respect to the ionization cross section as weight function (see Ref. 24 for details). It has been shown that for pulses of short duration only a small number of expansion terms has to be considered, which renders this scheme very efficient for femtosecond PP applications. ... 

It has long been considered that the use of non-orthogonal orbitals would lead to a formalism of immense complexity, which in turn would require computing resources that would make such an approach hopelessly inefficient. In fact we see that the opposite is true. The formalism leads to a description of molecules and chemical systems that is extremely compact and highly visual, and hence long expansions of the wave function in terms of different configurations, which obscures all our vital insight, are avoided. 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

In Section 2.1 we derived the expression for the transition rate kfi (2.22) by expanding the time-dependent wavefunction P(t) in terms of orthogonal and complete stationary wavefunctions Fa [see Equation (2.9)]. For bound-free transitions we proceed in the same way with the exception that the expansion functions for the nuclear part of the total wavefunction are continuum rather than bound-state wavefunctions. The definition and construction of the continuum basis belongs to the field of scattering theory (Wu and Ohmura 1962 Taylor 1972). In the following we present a short summary specialized to the linear triatomic molecule. 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

Both of Eqs. (21) and (22) involves N terms due to N permutations of the symmetric group SN, which is similar to a determinant expansion or a permanent, except for different coefficients. If one-electron functions are orthogonal, only a few terms are non-zero and make contributions to the matrix elements [42], and consequently the matrix elements are conveniently obtained. However, the use of... 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

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